# American Institute of Mathematical Sciences

2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

## Recurrence of multi-dimensional diffusion processes in Brownian environments

 1 Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan 2 Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
##### References:
 [1] T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206. [2] M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87. [3] K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441. [4] K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54. [5] K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965). [6] D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147. [7] G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45. [8] P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549. [9] Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256. [10] F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1. [11] H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171. [12] H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189. [13] H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.

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##### References:
 [1] T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206. [2] M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87. [3] K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441. [4] K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54. [5] K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965). [6] D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147. [7] G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45. [8] P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549. [9] Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256. [10] F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1. [11] H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171. [12] H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189. [13] H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.
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